QM Collision Exercise: Find Gamma Ray Wavelength in CMS

[Ibycus]

Homework Statement

A 2 MeV (kinetic energy) positron collides with an electron at rest. Find the wavelength of the resulting two gamma rays in the center of mass system. use the fact that $$E^{2} - p^{2}c^{2} = m^{2}c^{4}$$ is invariant between frames of reference for any system)

Homework Equations

As far as I know, I only really need:

$$E^{2} - p^{2}c^{2} = m^{2}c^{4}$$
The mass-energy of an electron: 511keV

The Attempt at a Solution

Using the conservation of energy equation given, it is easy to calculate the total energy for the gamma rays being equal to the mass-energy of each particle plus the kinetic energy, but I'm stuck at this point, because I don't know if it is possible to say how much energy goes to each gamma ray.

For the sake of symmetry I'd like to say that they each get exactly the same amount because to do otherwise would suggest some frame of reference in the collision has priority, but it seems an extremely shaky argument. All I'm sure of is the total energy. Could someone give me a nudge in the right direction?

This is also my first post on the physics forum, so if I'm asking for too much assistance my apologies, and I appreciate any help I might receive, so thanks!

 

[tiny-tim]

Welcome to PF!

Hi Ibycus! Welcome to PF!

… Find the wavelength of the resulting two gamma rays in the center of mass system.
…
For the sake of symmetry I'd like to say that they each get exactly the same amount because to do otherwise would suggest some frame of reference in the collision has priority, but it seems an extremely shaky argument.

Hint: in the centre of mass system, their total 3-momentum must be zero.